Topological invariants of Anosov representations

Abstract

We define new topological invariants for Anosov representations and study them in detail for maximal representations of the fundamental group of a closed oriented surface Σ into the symplectic group Sp (2n, R). In particular we show that the invariants distinguish connected components of the space of symplectic maximal representations other than Hitchin components. Since the invariants behave naturally with respect to the action of the mapping class group of Σ, we obtain from this the number of components of the quotient by the mapping class group action. For specific symplectic maximal representations we compute the invariants explicitly. This allows us to construct nice model representations in all connected components. The construction of model representations is of particular interest for Sp (4, R), because in this case there are −1−χ(Σ) connected components in which all representations are Zariski dense and no model representations have been known so far. Finally, we use the model representations to draw conclusions about the holonomy of symplectic maximal representations.

Maximal $${\mathsf {Sp}}(4,{\mathbb {R}})$$ surface group representations, minimal immersions and cyclic surfaces

“…Our approach builds on his work and provides the first mapping class group parameterization of what we call the Gothen components. For a closed oriented surface S of genus g ≥ 2, denote the associated character variety by X (Sp(4, R)), are Zariski dense, Guichard and Wienhard [15] have identified some special representations in these components, called hybrid representations, which are built by gluing together Fuchsian representation on subsurfaces of S. In Theorem 3.16, we give a mapping class group invariant parameterization of these 2 2g + 2g − 3 smooth connected components X d (Sp(4, R)) as fiber bundles over Teichmüller space. This is done by associating a preferred conformal structure to each representation ρ ∈ X d (Sp(4, R)) (Theorem 3.8).…”

Domains of discontinuity for surface groups

“…Using [8] we can show that the quotient space Ω ρ /ρ(π 1 (Σ)) is homeomorphic to an O(n)/O(n − 2)-bundle over the surface Σ.…”

Model Higgs Bundles in Exceptional Components of the Sp(4,R)-Character Variety

“…In [20], O. Guichard and A. Wienhard describe model maximal fundamental group representations ρ : π 1 (Σ) → Sp(4,R) in the components of R max . These models are distinguished into two subcategories, standard representations and hybrid representations.…”

“…The non-abelian Hodge theory correspondence provides a real-analytic isomorphism between the character variety R (G) and the moduli space M (G) of polystable G-Higgs bundles. The case when G = Sp(4,R) has received considerable attention by many authors, who studied the geometry and topology of the moduli space M (Sp(4,R)); see for instance [7], [10], [20]. The subspace of maximal Sp(4,R)-Higgs bundles, M max , that is, the one containing Higgs bundles with extremal Toledo invariant, has been shown to have 3 · 2 2g + 2g − 4 connected components [19].…”

“…From the point of view of the character variety R max , model representations in a subfamily of the 2g − 3 special components have been effectively constructed by O. Guichard and A. Wienhard in [20] by amalgamating certain fundamental group representations defined over topological surfaces with one boundary component.…”